Quadratic Functions; An Introduction Mr. J. Grossman

Graph the function y = x² Describe the graph. What does it look like?

Graph the function y = x² The graph is… Nonlinear U-shaped smooth curve (called a parabola) Symmetry Axis of Symmetry: the line that divides the parabola into two matching halves. It has a vertex.

Graph the function y = x² The function we have graphed and described is known as a Quadratic Function. A Quadratic Function is any function that can be written in the standard form: y = ax² + bx + c where a, b, and c are real numbers and a ≠ 0.

You must be able to recognize a Quadratic Function From a graph…

You must be able to recognize a Quadratic Function From a table of values… Constant change in x-values x y = x^2 1st Common Difference 2nd Common Difference 1 2 4 3 9 5 16 7

You must be able to recognize a Quadratic Function A quadratic function will not have a common first difference. It will have a common second difference. This is true of all quadratic functions. Constant change in x-values x y = x^2 1st Common Difference 2nd Common Difference 1 2 4 3 9 5 16 7

You must be able to recognize a Quadratic Function From an equation… Must be a 2nd degree polynomial y = 2x² y = -5x² + 6x f(x) = x² + 9 f(x) = ½ x² + 3x -11

Tell whether each function is a Quadratic Function: {(-4, 8), (-2, 2), (0, 0), (2, 2), (4, 8)} y = -3x + 20 y + 3x² = -4

Quadratic Functions Some open upwards; others, downward.

Quadratic Functions When a quadratic function is written in the form y = ax² + bx + c, the value of a determines the direction in which the parabola opens. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. Remember, a ≠ 0 !!!

Quadratic Functions Graph the function: f(x) = -4x² - x + 1 Does the parabola open upward or downward? Graph the function: y – 5x² = 2x – 6 Does the parabola open upward or downward?

Quadratic Functions The highest or lowest point on the parabola is known as the vertex. If a > 0, the parabola opens upward and the y-value of the vertex is the minimum value of the function. If a < 0, the parabola opens downward and the y-value of the vertex is the maximum value of the function.

Quadratic Functions

Quadratic Functions The value of a (coefficient of the x² term) affects the width of the parabola also. The smaller the absolute value of a, the wider the parabola. Compare the graphs of: y = -4x², y = ¼ x² + 3, and y = x². Which graph (quadratic function) is widest? Narrowest?

Quadratic Functions Recall that the quadratic function in standard form is written: y = ax² + bx + c. The value of c (constant of the quadratic function) translates the graph of the function up or down the axis of symmetry. Compare the graphs of the following quadratic functions: y = 2x², y = 2x² + 3, and y = 2x² - 3.

Quadratic Functions Positive values of c shift the vertex UP. Negative values of c shift the vertex DOWN.

Quadratic Functions DOMAIN: Unless a specific domain is given, you may assume that the domain of a quadratic function is the set of all real numbers. RANGE: The range begins (albeit minimum or maximum value) with the vertex.

Any questions? If not, complete Study Guide Practice 10-1, pg 128, all problems.